Vladimir Tarasenko, Paul Stastny and Troy Brouwer also scored for the Blues, who got their first regulation win in four tries against the Wild and improved to 8-3-1 in their last 12 at home.
Matt Dumba scored late in the second for the Wild, who lost their 10th game in their last 11.
Elliott improved to 7-3-2 since taking over for the injured Jake Allen (lower body), including making 23 saves in the second period.
Minnesota's 24 shots in the second period set a franchise road record for any period and was St. Louis' season high for shots given up in a period.
